
where the second step holds by exchanging the operations of the integration and the expectation, and the last step holds by using \({}_p{F_q}\left( {{\bf {AB}}} \right)={}_p{F_q}\left( {{\bf {A}}^{1/2}{\bf {B}}{\bf {A}}^{1/2}} \right)\)  for any positive definite matrix \(\bf A\)  and symmetric matrix \(\bf B\)  [1]}. Noticing that \(\bf Q\)  is an idempotent matrix with \(N_r-M+1\)  eigenvalues of value 1 and all other eigenvalues of value 0. More specifically, by defining \({\bf Y} = {{\bf {R}}_r}^{1/2}{{{\bf {\tilde{Z}}}}_w}\) , \(\bf Y\)  can be written as \({\bf Y}= {\bf {U\Sigma }}{{\bf {\Lambda }}^{\rm {H}}}\)  based on singular value decomposition, where \({\bf {\Sigma }}= {\rm diag}\left\lbrace \sqrt{{{\delta _i}}}\right\rbrace \in {\mathbb {R}}_{+}^{N_r\times (M-1)}\) , \({\bf U} \in {{\cal U}\left( {{N_r}} \right)}\) , \({\bf \Lambda } \in {{\cal U}\left( {{M-1}} \right)}\)  and \({{\cal U}\left( {{N}} \right)}\)  stands for the group of unitary \(N\times N\)  matrices. Accordingly, \(\bf Q\)  can be rewritten as
\({\bf {Q}} = {\bf {U}}\underbrace{\left( {{\bf {I}} - {\bf {\Sigma }}{{\left( {{{\bf {\Sigma }}^{\rm {H}}}{\bf {\Sigma }}} \right)}^{ - 1}}{{\bf {\Sigma }}^{\rm {H}}}} \right)}_{\bf {V}}{{\bf {U}}^{\rm {H}}},\) 
